Recent work on differential Galois theory

Let K be a differential field with an algebraically closed field of constants C of characteristic 0. To a linear differential equation of order n over K one associates a differential Galois group which is an algebraic subgroup of GL(n, C). The inverse problem reads: "which linear algebraic groups occur as differential Galois group ?" The inverse problem is solved by J.-P. Ramis for local and global analytic situations. A constructive solution of the inverse problem for the differential field C(x) (for connected groups) has been given by C. Mitschi and M.F. Singer.